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SarkariResultOutBMS · 2024

Previous Year Question

The Fourier transform X(jω) of x(t) = e^(−2|t − 1|) is
  1. 4e^(−jω)/(4 + ω²)
  2. e^(−jω)/(4 + ω²)
  3. 4e^(jω)/(4 + ω²)
  4. e^(jω)/(4 + ω²)

Correct answer: A

Explanation

Use the pair e^{-a|t|}↔2a/(a²+ω²). For a=2 the base transform is 4/(4+ω²). Replacing t by t−1 shifts the signal right by 1 s and multiplies the spectrum by e^{-jω}. Thus X(jω)=4e^{-jω}/(4+ω²), option A.

Formula

e^{-2|t−1|} ↔ 4e^{-jω}/(4+ω²)

Step-by-step solution

Shifted Two-Sided Exponential explanatory diagram

Correct Answer

A — 4e^(−jω)/(4 + ω²)

Concept and Reasoning

Use the pair e^{-a|t|}↔2a/(a²+ω²). For a=2 the base transform is 4/(4+ω²). Replacing t by t−1 shifts the signal right by 1 s and multiplies the spectrum by e^{-jω}. Thus X(jω)=4e^{-jω}/(4+ω²), option A.

Calculation / Key Relation

e^{-2|t−1|} ↔ 4e^{-jω}/(4+ω²)

Exam Takeaway

Right time shift → negative spectral phase factor e^{-jωt0}.

Common Mistake

Do not change the magnitude denominator because of the time shift.

Common mistake

Do not change the magnitude denominator because of the time shift.

Exam tip

Right time shift → negative spectral phase factor e^{-jωt0}.

Quick method

Right time shift → negative spectral phase factor e^{-jωt0}.